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Monday, June 3, 2013

We really have no way of knowing what
early humans thought when they gazed up at the sky. Since everyday
practical experience is, by definition, limited to a very small
region of space and time, it requires considerable cognitive
sophistication to conceive of something – say the night sky –
“going on for ever,” let alone to ponder whether that means it is
“infinite,” or indeed what “infinite” actually means.

What we do know is that the ancient
Greeks made what may have been the first substantial attempt to
analyze the notion of infinity, with Zeno of Elea (ca 490-430 BCE) of
particular note for his discussion of a number of (seeming) paradoxes
that arise from the assumption that space and time are (or are not)
infinitely divisible.

Archimedes’ (ca 287-112 BCE)
calculations of areas and volumes made implicit use
of infinity, and from today’s perspective can be recognized as the
forerunner of integral calculus.

Skillful formal –
though by modern standards not rigorous – use of the infinitely
large and the infinitely small was made by Isaac Newton and Gottfried
Leibniz in their development of modern infinitesimal calculus in the seventeenth century, though it was not until the nineteenth century
when Bernard
Bolzano, Augustin-Louis
Cauchy, and Karl
Weierstrass finessed the lurking problems of infinity by means of
the famous (and for many a first-year mathematics major, infamous)
epsilon-delta definitions of limits and continuity.

But none of these
developments was about infinity as an entity;
the focus rather was on the unending nature of certain processes,
starting with counting. It wasGeorg
Cantor (1845 – 1918) who really tackled infinity head on. His
proof that the set of real numbers cannot be put into one-one
correspondence with the natural numbers, and hence is of a larger
order of infinitude, led to a series of papers, published in a
remarkable ten-year period between 1874 and 1884, that formed the
basis for modern abstract set theory, including the development of a
fully formed arithmetical theory of infinite numbers (or
“cardinals”).

Reactions to Cantor’s revolutionary
new ideas ranged from outraged condemnation to fulsome praise.
Henri Poincaré called Cantor’s work a “grave disease”
that threatened to infect mathematics, and Leopold Kronecker
described Cantor as a “scientific charlatan” and a “corrupter
of youth.” Ludwig Wittgenstein, writing long after Cantor's death,
complained that mathematics had become “ridden through and through
with the pernicious idioms of set theory,” a theory he dismissed as
“utter nonsense,” “laughable,” and “wrong.”

At the other end
of the spectrum, in 1904, in the UK the Royal
Society awarded Cantor its highest award, the Sylvester
Medal, and in Germany David
Hilbert declared that “No one shall expel us from the Paradise
that Cantor has created.”

Having devoted
the early part of my professional career to work in (infinitary) set
theory, starting with my Ph.D. in “large cardinal theory,”
completed in 1971, and moving on to work on alternative universes of
sets (a particularly hot topic after Paul Cohen’s introduction of
the method of forcingin 1963), in
the early 1980s my interests started to shift elsewhere, to questions
about information, communication, and human reasoning.

Both discussions
raised the question as to whether study of infinity – in particular
the hierarchy of larger infinities that Cantor bequeathed to us –
would ever have any practical applications. As panelist Hugh Woodin
remarked at one point in the discussion, it is a foolish
mathematician who declares that a particular piece of mathematics
will not find applications. For instance, G. H. Hardy’s famous
statement (in his book A
Mathematician’s Apology) that his
work in number theory would never find practical application, proved
to be spectacularly wrong less than a century later, when number
theory became the foundation for internet security.

Hardy’s
observation was based on his familiarity of the world he lived in, a
world in which the World Wide Web was not even a dream. Today, we
cannot know what the world of tomorrow will look like. On the other
hand, whatever our children and grandchildren will take for granted,
their world will surely be finite, which makes it unlikely that
Cantor’s theory – and the almost a century of development in set
theory since then – will have practical use.

Or does it? What
about calculus? Infinitesimal (!) calculus not only has applications
in the modern world, but much of the science, technology, medicine,
and even financial structure the underpins our world depends on
calculus for its very existence. Applications don’t get more real
than that.

True, but the
dependence on infinities you find in calculus is essentially
asymptotic. What really drives calculus is the unending nature of
certain processes on the natural numbers. Talk of “infinitely
large” or “infinitely small” is little more than a manner of
speaking. Indeed, the epsilon-delta definitions (which do not involve
infinities or infinitesimals) are a way to formalize that manner of
speaking, effectively eliminating any actual infinite or
infinitesimal quantities.

In contrast, much
of the work on infinity (more precisely, infinities) carried out in
the second half of the twentieth century (when I was working in that
area) focused on properties of sets that made their cardinalities
super-infinities of different orders: inaccessible cardinals, Ramsey
cardinals, measurable cardinals, compact cardinals, supercompact
cardinals, Woodin cardinals, and so on. Infinities which dwarfed into
invisibility the puny cardinality of the set of natural numbers.
Indeed, each one in that sequence dwarfed all its predecessors into
invisibility. How could that work find an application?

I’ll lay my
cards on the table. I think the chances are that it won’t. But I don’t
think it is impossible. Indeed, I began to suspect a possible
application in the very domain I worked in after I left set theory.

[This may of
course be nothing more than a reflection of having at my disposal a
large hammer which made everything look a bit like a nail. But let’s
press on.]

The post 9/11
world saw me involved in a series of Defense Department projects the
first being improving intelligence analysis (and the others
essentially variants of that).

In today’s
information rich world, the major nations can be assumed to have
access to all the information they need to predict (and hopefully
thence prevent) the majority of terrorist attacks. The trouble is,
the few data points which must be identified and connected together
to determine the likelihood of a future attack are just a tiny few in
an overwhelming ocean of data. Even in the era of cloud computing,
identifying the key information is analogous to using the naked eye
to find a handful of proverbial needles in a non-proverbial field of
haystacks.

To all intents
and purposes, the available data is infinite. The only hope is to
impose some structure on the data that makes it possible for humans
and computers to work together on it, narrowing down the focus to the
regions more likely to be of significance. Though modern computing
systems can sift through massive (finite) amounts of data in a
relatively short time, they need to be programmed, and writing those
programs (at least, some kinds of them) will require some structure
on those large sets of data. A possible place to find the appropriate
structure(s) is infinitary set theory. In other words, to develop the
appropriate structures, assume the data is infinite. View the
infinite as a theoretical simplification of the very large finite.
(Economists sometimes make a similar simplifying assumption about
economies.)

Do I think this
is likely? Frankly, no. But then, neither could Hardy conceive of any
practical application of his work in number theory. [Incidentally,
like Hardy, I don’t think mathematics needs applications to justify
itself. It’s just that the question of application is what this
article is about!]

The discussion
about large cardinals you will find in those panel discussions at the
World Science Festival might seem impossibly abstract and far removed
from the everyday world. Indeed, it is. But the questions being
discussed all resulted from a process of rigorous, logical
investigation that arose directly from late nineteenth century
attempts to understand heat flow. History tells us that what begins
in the real world, very often ends up being used in the real world.

Prediction is
hard, particularly about the future.

Incidentally, how
did I end up working on a project for the DoD? They asked me. I might
not be the only person to speculate about a possible use of Cantor’s
paradise. This is your taxpayer dollars at work.

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The Mathematical Association of America is the world's largest community of mathematicians, students, and enthusiasts. We accelerate the understanding of our world through mathematics, because mathematics drives society and shapes our lives. Visit us at maa.org.